A runs test for stock-market prices with an unobserved trend

To test whether stock-market prices follow a random walk, the algebraic signs of their returns have been analyzed like a sequence of coin tosses. Similar to coin tosses, which represent Bernoulli trials with equiprobable outcomes, signed returns lend themselves for a simple runs test for randomness. However, stock-market prices typically comprise an unobserved trend implying that even under pure randomness positive and negative returns are not necessarily equally probable. Therefore, it is difficult to infer the behavior of stock-market prices from the traditional runs test. Fortunately, the von Neumann algorithm allows to transform tosses of a potentially unfair coin, meaning that it could suffer from an unknown bias, into equiprobable outcomes. This is done by tossing the coin twice and retaining only the first observation when different outcomes arise. Applying the same “trick” to pairs across a sequence of signed stock-market returns paves the way for a runs test that is robust to the effects of constant, unobserved trends.